Optimal Control, Construction, and Analysis of λ-Bernstein Bézier Surfaces for Quasi-Harmonic Functional
Abstract
In this article, we investigate a novel construction scheme for λ-Bernstein B´ ezier surfaces and illustrate it with biquadratic and bicubic cases to demonstrate their geometric characteristics and their applications. The primary objective is to investigate how the shape parameter λ improves control over surface smoothness and facilitates an optimal solution in surface design. We examine the geometric properties of these surfaces, including mean and Gaussian curvature, shape operator coefficients, and GaussWeingarten coefficients. We also analyze the extremal conditions for λ-Bernstein B´ezier surfaces derived from the vanishing condition for the gradient of the quasi-harmonic functional. Integral formulations based on Bernstein polynomials enable precise computation of the vanishing gradient condition, allowing us to determine the constraints on interior control points in terms of known boundary control points. Graphical illustrations validate the approach by providing a better understanding of the geometric properties of these surfaces, including improved surface smoothness and design flexibility. They effectively showcase the behavior of λ-Bernstein polynomials and their corresponding surfaces. Computational results demonstrate the effectiveness of this method for applications in computer graphics, computational geometry, computer science, and engineering, offering a robust framework for analyzing and generating optimal surfaces, contributing to advancements across various scientific disciplines.
References
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S. Araci, M. Acikgoz, and E. S ̧en, “On the extendedKimsp-adicq-deformed fermionicintegrals in thep-adic integer ring,”Journal of Number Theory, vol. 133, no. 10, pp. 3348– 3361, 2013. doi: 10.1016/j.jnt.2013.04.007
S. Araci and M. Acikgoz, “A note on the values of the weightedq-Bernstein polynomi-als and weightedq-Genocchi numbers,”Advances in Difference Equations, 01 2015. doi:10.1186/s13662-015-0369-y
D. S. K. Lee-Chae Jang, Taekyun Kim and D. V. Dolgy, “Onp-adic fermionic integralsofq-Bernstein polynomials associated withq-Euler numbers and polynomials,”Journal ofNumber Theory, vol. 1, no. 10, pp. 1–9, 2018. doi: 10.1016/j.jnt.2013.04.007
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N. Turhan, F. ̈Ozger, and M. Mursaleen, “Kantorovich-Stancu type (α,λ,s) - Bernsteinoperators and their approximation properties,”Mathematical and Computer Modelling ofDynamical Systems, vol. 30, no. 1, pp. 228–265, 2024. doi: 10.1080/13873954.2024.2335382
Z.-P. Lin, G. Torun, E. Kangal, ̈U. D. Kantar, and Q.-B. Cai, “On the properties of themodifiedλ-Bernstein-Stancu operators,”Symmetry, vol. 16, no. 10, p. 1276, 2024. doi:10.3390/sym16101276
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C. Zhu, P. Gu, D. Liu, X. Hu, and Y. Wu, “Evaluation of surface topography of SiCp/Alcomposite in grinding,”The International Journal of Advanced Manufacturing Technology,vol. 102, pp. 2807–2821, 2019.
J. Zhang and J. Wang, “Research on the effect path of institutional environment to internalcontrol,” in7th International Conference on Social Science and Higher Education (ICSSHE2021). Atlantis Press, 2021, pp. 664–668.
P. Gu, J. Chen, W. Huang, Z. Shi, X. Zhang, and L. Zhu, “Evaluation of surface quality anderror compensation for optical aspherical surface grinding,”Journal of Materials ProcessingTechnology, vol. 327, p. 118363, 2024. doi: 10.1016/j.jmatprotec.2024.118363
A. Goetz,Introduction to Differential Geometry.Addison Wesley Publishing Company,1970.
M. P. do Carmo,Differential Geometry of Curves and Surfaces. Prentice Hall, 1976.
F. C. Park, “Optimal Robot Design and Differential Geometry,”Journal of MechanicalDesign, vol. 117, no. B, pp. 87–92, 06 1995. doi: 10.1115/1.2836475
N. Jaquier and T. Asfour, “Riemannian geometry as a unifying theory for robot motionlearning and control,” inRobotics Research, A. Billard, T. Asfour, and O. Khatib, Eds.Cham: Springer Nature Switzerland, 2023. ISBN 978-3-031-25555-7 pp. 395–403.
S. Radhakrishnan and W. Gueaieb, “A state-of-the-art review on topology and differentialgeometry-based robotic path planning-part I: planning under static constraints,”Interna-tional Journal of Intelligent Robotics and Applications, 03 2024. doi: 10.1007/s41315-024-00330-5
A. Ai, M. Ifrim, and D. Tataru, “The time-like minimal surface equation in Minkowskispace: low regularity solutions,”Inventiones mathematicae, vol. 235, no. 3, pp. 745–891,March 2024. doi: 10.1007/s00222-023-01231-3
J. Monterde and H. Ugail, “On harmonic and biharmonic B ́ezier surfaces,”Computer AidedGeometric Design, vol. 21, no. 7, pp. 697 – 715, 2004. doi: 10.1016/j.cagd.2004.07.003
D. Ahmad and B. Masud, “A Coons patch spanning a finite number of curves tested forvariationally minimizing its area,”Abstract and Applied Analysis, vol. 2013, 2013. doi:10.1155/2013/645368
G.Xu, R. Timon, G. Erhan, W. Qing, K. Hui, and W. Guozhao, “Quasi-harmonic B ́ezierapproximation of minimal surfaces for finding forms of structural membranes,”Comput.Struct., vol. 161, no. C, pp. 55–63, Dec. 2015. doi: 10.1016/j.compstruc.2015.09.002
D. Ahmad and B. Masud, “Variational minimization on string-rearrangement surfaces, illus-trated by an analysis of the bilinear interpolation,”Applied Mathematics and Computation,vol. 233, pp. 72 – 84, 2014. doi: 10.1016/j.amc.2014.01.172
D. Ahmad and S. Naeem, “Quasi-harmonic constraints for toric B ́ezier surfaces,”SigmaJournal of Engineering and Natural Sciences, vol. 36, pp. 325–340, 2018.
D. Ahmad, S. Naeem, A. Haseeb, and M. Mahmood, “A computational approach to aquasi- minimal B ́ezier surface for computer graphics,”VFAST Transactions on SoftwareEngineering, vol. 9, no. 4, pp. 150–159, 2021. doi: 10.21015/vtse.v9i4.929
D. Ahmad and B. Masud, “Near-stability of a quasi-minimal surface indicated through atested curvature algorithm,”Computers & Mathematics with Applications, vol. 69, no. 10,pp. 1242 – 1262, 2015. doi: 10.1016/j.camwa.2015.03.015
J. Hoschek and D. Lasser,Fundamentals of Computer Aided Geometric Design.A KPeters/CRC Press, 2003.
D. Marsh,Applied Geometry for Computer Graphics and CAD. Springer Science & Busi-ness Media, 2005.
G. Farin,Curves and Surfaces for Computer-Aided Geometric Design: A PracticalGuide, ser. Computer science and scientific computing.Elsevier Science, 2014. ISBN9781483296999
X. Jiao, D. Wang, and H. Zha, “Simple and effective variational optimization of surface andvolume triangulations,”Engineering with Computers, vol. 27, pp. 81–94, 2011.
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt,Exact Solutions ofEinstein’s Field Equations, ser. Cambridge Monographs on Mathematical Physics. Cam-bridge University Press, 2003. ISBN 9781139435024
D. Ahmad and M. Ziad, “Homothetic motions of spherically symmetric space-times,”Jour-nal of Mathematical Physics, vol. 38, no. 5, pp. 2547–2552, 1997. doi: 10.1063/1.531994
D. Ahmad and K. Habib, “Homotheties of a class of spherically symmetric space-timeadmittingG3as maximal isometry group,”Advances in Mathematical Physics, vol. 2018,p. 8195208, Jan 2018. doi: 10.1155/2018/8195208
U. Camci and K. Saifullah, “Conformal symmetries of the energy–momentum tensor ofspherically symmetric static spacetimes,”Symmetry, vol. 14, no. 4, p. 647, Mar 2022. doi:10.3390/sym14040647
K. S. Nisar, J. Ali, M. K. Mahmood, D. Ahmad, and S. Ali, “Hybrid evolutionaryPad ́e approximation approach for numerical treatment of nonlinear partial differentialequations,”Alexandria Engineering Journal, vol. 60, no. 5, pp. 4411–4421, 2021. doi:10.1016/j.aej.2021.03.030
A. Zafar, M. Ijaz, A. Qaisar, D. Ahmad, and A. Bekir, “On assorted soliton wave solu-tions with the higher-order fractional Boussinesq-Burgers system,”International Journal ofModern Physics B, vol. 0, no. 0, p. 2350287, 0. doi: 10.1142/S0217979223502879
M. Saqib, D. Ahmad, A. N. Al-Kenani, and T. Allahviranloo, “Fourth- and fifth-orderiterative schemes for nonlinear equations in coupled systems: A novel adomian decom-position approach,”Alexandria Engineering Journal, vol. 74, pp. 751–760, 2023. doi:10.1016/j.aej.2023.05.047
C. Cos ́ın and J. Monterde,B ́ezier surfaces of minimal area. Berlin, Heidelberg: SpringerBerlin Heidelberg, 2002, pp. 72–81. ISBN 978-3-540-46080-0
J. Monterde, “B ́ezier surfaces of minimal area: The Dirichlet approach,”Computer AidedGeometric Design, vol. 21, no. 2, pp. 117 – 136, 2004. doi: 10.1016/j.cagd.2003.07.009
J. Monterde and H. Ugail, “A general 4th-order{PDE}method to generate B ́ezier surfacesfrom the boundary,”Computer Aided Geometric Design, vol. 23, no. 2, pp. 208 – 225, 2006.doi: 10.1016/j.cagd.2005.09.001
L.-Y. Sun and C.-G. Z. (2016), “Approximation of minimal toricB ́ezier patch,”Advancesin Mechanical Engineering, vol. 8, p. (6).
D. Ahmad, K. Hassan, M. K. Mahmood, J. Ali, I. Khan, and M. Fayz-Al-Asad, “Variation-ally improved B ́ezier surfaces with shifted knots,”Advances in Mathematical Physics, vol.2021, no. 5, p. 14, 2021. doi: 10.1155/2021/9978633
Y. X. Hao and T. Li, “Construction of quasi-B ́ezier surfaces from boundary conditions,”Graphical Models, vol. 123, p. 101159, 2022. doi: 10.1016/j.gmod.2022.101159
X. Li, Y. Zhu, H. Wu, J. Xu, R. Ling, X. Wu, and G. Xu, “Construction of B ́ezier surfaceswith energy-minimizing diagonal curves from given boundary,”Journal of Computationaland Applied Mathematics, vol. 413, p. 114382, 2022. doi: 10.1016/j.cam.2022.114382
D. Ahmad, M. K. Mahmood, Q. Xin, F. M. O. Tawfiq, S. Bashir, and A. Khalid, “A com-putational model forq-BernsteinQuasi-minimal B ́ezier surface,”Journal of Mathematics,vol. 2022, p. 8994112, Sep 2022.
D. Ahmad, K. Naz, M. E. Buttar, P. C. Darab, and M. Sallah, “Extremal solutions forsurface energy minimization: Bicubically blended Coons patches,”Symmetry, vol. 15, no. 6, 2023. doi: 10.3390/sym15061237
S. Bashir and D. Ahmad, “Geometric analysis of non-degenerate shifted-knots B ́ezier sur-faces in minkowski space,”PLOS ONE, vol. 19, no. 1, pp. 1–28, 01 2024. doi: 10.1371/jour-nal.pone.0296365
S. Bernstein, “D ́emonstration du th ́eor`eme de Weierstrass fond ́ee sur le calcul desprobabilit ́es,”Comm. Soc. Math. Kharkov, vol. 13, pp. 1–2, 1912. [Online]. Available:http://mi.mathnet.ru/khmo107
K. Khan and D. Lobiyal, “B ́ezier curves based on Lupa ̧s (p,q)-analogue of Bernstein func-tions in CAGD,”Journal of Computational and Applied Mathematics, vol. 317, pp. 458–477,2017. doi: 10.1016/j.cam.2016.12.016
G. M. Phillips, “A generalization of the Bernstein polynomials based on theq-integers,”The ANZIAM Journal, vol. 42, no. 1, pp. 79–86, 07 2000. doi: 10.1017/S1446181100011615
H. Oruc and G. M. Phillips, “q-Bernstein polynomials and B ́ezier curves,”Journal of Com-putational and Applied Mathematics, vol. 151, no. 1, pp. 1 – 12, 2003. doi: 10.1016/S0377-0427(02)00733-1
D. Kim, L.-C. Jang, Y. Kim, and J. Choi, “Onp-adic analogue ofq-Bernstein polynomialsand related integrals,”Discrete Dynamics in Nature and Society, vol. 2010, 09 2010. doi:10.1155/2010/179430
T. Kim, “A note onq-Bernstein polynomials,”Russian Journal of Mathematical Physics,vol. 18, no. 1, p. 73, 2011. doi: 10.1134/S1061920811010080
S. Araci, M. Acikgoz, and E. S ̧en, “On the extendedKimsp-adicq-deformed fermionicintegrals in thep-adic integer ring,”Journal of Number Theory, vol. 133, no. 10, pp. 3348– 3361, 2013. doi: 10.1016/j.jnt.2013.04.007
S. Araci and M. Acikgoz, “A note on the values of the weightedq-Bernstein polynomi-als and weightedq-Genocchi numbers,”Advances in Difference Equations, 01 2015. doi:10.1186/s13662-015-0369-y
D. S. K. Lee-Chae Jang, Taekyun Kim and D. V. Dolgy, “Onp-adic fermionic integralsofq-Bernstein polynomials associated withq-Euler numbers and polynomials,”Journal ofNumber Theory, vol. 1, no. 10, pp. 1–9, 2018. doi: 10.1016/j.jnt.2013.04.007
K. Khan and D. Lobiyal, “B ́ezier curves based on lupa ̧s (p,q)-analogue of Bernstein functionsin CAGD,”Journal of Computational and Applied Mathematics, vol. 317, pp. 458–477,2017. doi: 10.1016/j.cam.2016.12.016
t.A ̆gy ̈uz and t.A ̧cikg ̈oz, “A note on Bernstein polynomials based on (p,q)-calculus,”AIPConference Proceedings, vol. 1978, no. 1, p. 040006, 2018. doi: 10.1063/1.5043690
Q.-B. Cai, B.-Y. Lian, and G. Zhou, “Approximation properties ofλ-Bernstein operators,”Journal of Inequalities and Applications, vol. 2018, no. 1, p. 61, 2018. doi: 10.1186/s13660-018-1653-7
M. Ayman-Mursaleen, M. Nasiruzzaman, N. Rao, M. Dilshad, and K. S. Nisar, “Approxi-mation by the modifiedλ-Bernstein-polynomial in terms of basis function,”AIMS Mathe-matics, vol. 9, no. 2, pp. 4409–4426, 2024. doi: 10.3934/math.2024217
J. Delgado and J. Pe ̃na, “Geometric properties and algorithms for rationalq-B ́ezier curvesand surfaces,”Mathematics, vol. 8, no. 4, p. 541, 2020. doi: 10.3390/math8040541
G. Zhou and Q.-B. Cai, “Bivariateλ-Bernstein operators on triangular domain,”AIMSMathematics, vol. 9, no. 6, pp. 14 405–14 424, 2024. doi: 10.3934/math.2024700
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Published
2025-02-05
How to Cite
BUTTAR, Mariyam Ehsan; AHMAD, Daud.
Optimal Control, Construction, and Analysis of λ-Bernstein Bézier Surfaces for Quasi-Harmonic Functional.
Yugoslav Journal of Operations Research, [S.l.], feb. 2025.
ISSN 2334-6043.
Available at: <https://yujor.fon.bg.ac.rs/index.php/yujor/article/view/1320>. Date accessed: 13 mar. 2025.
doi: https://doi.org/10.2298/YJOR240815051B.
Issue
Section
Research Articles
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
